Probabilistic circuits (PCs) represent a probability distribution as computational graph. Enforcing structural properties on these graphs guarantees that several inference scenarios become tractable. Among properties, structured decomposability is particularly appealing one: it enables the efficient and exact computations of complex logical formulas, can be used to reason about expected output ...

A subset S of {0, 1, . . . , 2t − 1}n is called a t-fold MDS code if every line in each of n base directions contains exactly t elements of S. The adjacency graph of a t-fold MDS code is not connected if and only if the characteristic function of the code is the repetition-free sum of the characteristic functions of t-fold MDS codes of smaller lengths. In the case t = 2, the theory has the foll...

Let G be a connected graph on n vertices, and let ; ; and be edge-disjoint cycles in G such that (i) ; (resp. ;) are vertex-disjoint and (ii) jj + jj = jj+ jj = n, where jj denotes the length of. We say that ; ; and yield two edge-disjoint hamiltonian cycles by edge exchanges if the four cycles respectively contain edges e; f; g and h such that each of (? feg) S (? ffg) S fg; hg and (? fgg) S (...

The critical group of a connected graph is a finite abelian group, whose order is the number of spanning trees in the graph. The structure of this group is a subtle isomorphism invariant that has received much attention recently, partly due to its relation to the graph Laplacian and chip-firing games. However, the group structure has been determined for relatively few classes of graphs. We conj...

Many AI tasks can be formalized as constraint satisfaction problems (CSPs), which involve finding values for variables subject to constraints. 'vfrhile solving a CSP is an NP-coiapktte task in geneid, tractable classes of CSPs have been identified based on the structure of the underlying constraint graphs. Much effort has been spent on exploiting structural properties of the constraint graph to...

The L-decomposable and the bi-decomposable models are two families of distributions on the set Sn of all permutations of the first n positive integers. Both of these models are characterized by collections of conditional independence relations. We first compute a Markov basis for the L-decomposable model, then give partial results about the Markov basis of the bi-decomposable model. Using these...

We characterize the graphs that admit a decomposition into circuits, i.e. finite or infinite connected 2-regular graphs. Moreover, we show that, as is the case for the removal of a closed eulerian subgraph from a finite graph, removal of a non-dominated eulerian subgraph from a (finite or infinite) graph does not change its circuit-decomposability or circuit-indecomposability. For cycle-decompo...

‎recently‎, ‎some techniques such as adding whiskers and attaching graphs to vertices of a given graph‎, ‎have been proposed for constructing a new vertex decomposable graph‎. ‎in this paper‎, ‎we present a new method for constructing vertex decomposable graphs‎. ‎then we use this construction to generalize the result due to cook and nagel‎.

The Matching-Cut problem is the problem to decide whether a graph has an edge cut that is also a matching. Chvátal studied this problem under the name of the Decomposable Graph Recognition problem, and proved the problem to be NP-complete for graphs with maximum degree 4, and gave a polynomial algorithm for graphs with maximum degree 3. Recently, unaware of Chvátal’s result, Patrignani and Pizz...